Fig. 1. Theoretical model of the two-coupled-oscillator system.

Fig. 2. (a) Schematic of the bilayer grating for a freely designed BIC. (b) Dispersion relation of single layer grating. (c) Band structure of the bilayer grating at d/a=0.99. The red circle represents the position of kxa/2π=0.122. (d) The corresponding Q factors. (e) Far-field polarization states around BIC. (f) The dependence of BICs on the interlayer distance d and kxa/2π. The inset gives Q factors around the four BIC modes (ω+) for four different interlayer distances indicated by the color points.

Fig. 3. (a) Simulated band structure at d/a=0.68. (b) Calculated polarization vectors around BICs with the Q factors as background in the Brillouin zone for d/a=0.68. (c) Sign of polarization vector components around BICs. (d) Calculated polarization vectors around BICs with the Q factors as background in the Brillouin zone for d/a=0.39.

Fig. 4. Schematic view and photos of the experimental setup and sample.

Fig. 5. Simulated transmittance spectra of the structure for TM polarization at (a) d/a=0.99 and (d) d/a=1.23. The circles represent the positions of BICs. (b), (e) Corresponding Stokes parameters around BICs points. (c), (f) Corresponding experimental results.

Fig. 6. Calculated polarization vectors around BICs in momentum space based on the eigenfield extraction method for (a) d/a=0.99 and (c) d/a=1.23. (b), (d) Corresponding results based on the transmission spectra extraction method.

Fig. 7. Schematic of the bilayer grating structure for highly efficient SHG. (b) Reflectance spectra of the structure at different incident angles for TM polarization. (c) Electric field distributions at the x–z cross section (y=a/2) and the y–z cross section (x=a/2) at the wavelength of quasi-BIC at θ=9°. (d) Calculated reflectance and SHG conversion efficiency versus the wavelength of pumping light at θ=8.3°. (e) Dependence of the SHG conversion efficiency on the incident angle. (f) Dependence of the SHG conversion efficiency on the interlayer distance at θ=8.9°.

Fig. 8. Reflectance spectrum of the single grating for TM polarization at kxa/2π=0.122. Red solid line represents the calculated reflectance spectra. Blue circle denotes the fitting reflectance spectrum based on SRM.

Fig. 9. Schematic of a system with two-fold in-plane rotational symmetry and mirror-symmetry along z.

Fig. 10. Schematic illustration for the relevant angles and vectors in determining the polarization vector and its projection onto the x–y plane.